src/HOL/Arith.thy
 author paulson Fri Jun 14 12:22:59 1996 +0200 (1996-06-14) changeset 1796 c42db9ab8728 parent 1475 7f5a4cd08209 child 1824 44254696843a permissions -rw-r--r--
Tidied spacing
1 (*  Title:      HOL/Arith.thy
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
4     Copyright   1993  University of Cambridge
6 Arithmetic operators and their definitions
7 *)
9 Arith = Nat +
11 instance
12   nat :: {plus, minus, times}
14 consts
15   pred      :: nat => nat
16   div, mod  :: [nat, nat] => nat  (infixl 70)
18 defs
19   pred_def  "pred(m) == nat_rec m 0 (%n r.n)"
20   add_def   "m+n == nat_rec m n (%u v. Suc(v))"
21   diff_def  "m-n == nat_rec n m (%u v. pred(v))"
22   mult_def  "m*n == nat_rec m 0 (%u v. n + v)"
24   mod_def   "m mod n == wfrec (trancl pred_nat)
25                           (%f j. if j<n then j else f (j-n)) m"
26   div_def   "m div n == wfrec (trancl pred_nat)
27                           (%f j. if j<n then 0 else Suc (f (j-n))) m"
28 end
30 (*"Difference" is subtraction of natural numbers.
31   There are no negative numbers; we have
32      m - n = 0  iff  m<=n   and     m - n = Suc(k) iff m)n.
33   Also, nat_rec(m, 0, %z w.z) is pred(m).   *)